Download PDF by Duan Li Xiaoling Sun: Nonlinear Integer Programming (International Series in

By Duan Li Xiaoling Sun

ISBN-10: 0387295038

ISBN-13: 9780387295039

ISBN-10: 0387329951

ISBN-13: 9780387329956

A mix of either Integer Programming and Nonlinear Optimization, it is a strong booklet that surveys the sector and gives a cutting-edge therapy of Nonlinear Integer Programming. it's the first publication on hand at the topic. The booklet goals to deliver the theoretical starting place and resolution equipment for nonlinear integer programming to scholars and researchers in optimization, operations study, and laptop technological know-how.

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11. D If hj{x) > 0 for any x G X, j = 1 , . . 1 General case We consider the following unconstrained binary quadratic optimization problem: (BQ) min a:G{-l,l}"' q(x) = -x^Qx + b^x^ 2 Optimality, Relaxation and General Solution Procedures where Q is a symmetric matrix in R^^^ and 6 G M^. , n, can be transformed into the form of {BQ) by the Unear transformation: y^ — l^ + [m — li){xi + l)/2, i = 1 , . . , n. t. X;^^ — 1, Z = 1, . . , n . Problem {CQ) is essentially a nonconvex continuous optimization problem even if matrix Q is positive semidefinite.

So Az"" < b. Moreover, since c^d'^ = -XjAid^ < 0 for alH = 1 , , . 6) that c^^:* < c^z. 3), Moreover, combining c^2:* < c^z with the optimality of z and (2,4,6) leads to c^d'^ = 0 for z with /i^ > 1. 4,8) Then, Aix* = Aix - [fii\Aid^ [fit\Aid^ < Aix < b\ (2,4,9) Also, by (2,4,5), it holds X* - ^ + (/ii - [fli\)d^ + ... + (^^- [fit\)d\ Thus, we obtain ||x* — ^||oo < nA{A) using the similar arguments as in part (i). Moreover, we have A2X'' - A2Z + {in - [iii\)A2S + • • • + (/i^ - [iit\)A2d^ < A2Z < b^.

D Optimality, Relaxation and General Solution Procedures Notice that problem {BQ) is a box constrained convex quadratic programming problem and hence is much easier to solve than (BQ). Solving {BQ), however, in general only yields a real solution. The next result gives a sufficient condition for getting a nearby integer optimal solution to (BQ) based on a real optimal solution to (BQ). 15 Assume that Q is a real positive semidefinite matrix and X* is an optimal solution to {BQ). If z"" e. { — 1, l}'^ satisfies the following conditions: (i) zl = x^forxl e {-1,1}, and (ii) 2'*(5(2:* - X*) < Xmin{Q)e, where Z* = diag{z'') and \rnin{Q) is the minimum eigenvalue ofQ, then 2:* is an optimal solution to {BQ).

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Nonlinear Integer Programming (International Series in Operations Research & Management Science) by Duan Li Xiaoling Sun

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